Correspondència referida a l'article de R. Giannetti, publicat ibid. vol.49 p.87-88Peer Reviewe

Pallàs Areny, Ramon

Vargas Drechsler, Manuel

IEEE Transactions on Instrumentation and Measurement

English

Pallàs-Areny, Ramon

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398 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 51, NO. 2, APRIL 2002Correspondence________________________________________________________________________Comments on “On Resistor-Induced Thermal Noise inLinear Circuits”Manuel Vargas and Ramon Pallàs-ArenyIn a recent paper [1], Giannetti considered the thermal noise gen-erated by a resistor R on a generic output port in any linear network.He reasoned that, because the contribution of the thermal noise of theresistor to the output noise of the circuit vanishes for both R! 0 andR ! 1, there should be a resistor value whose contribution to theoutput noise is maximal. In order to find that value, from Fig. 1 he ob-tained that the thermal noise contributed by R was [1, (3)]v2u(R) = jH 0j2 RR+ Z24kTR= ARR+ Z21R(1)which is incorrect because the second and third terms of (1) do not havethe dimensions of voltage spectral density. The correct analysis yieldsv2u(R) = jH 0j2 RZR+ Z24kTR= ARZR+ Z21R: (2)The maximal noise will result for that value of R that nulls the firstderivative of (2), which is R = jZj. This agrees with [1, (5)] becausethe mistake in (1) reduces to a multiplying factor (Z) that does notdepend on R. The maximal output noise contributed by the thermalnoise of R can then be calculated by replacing R with jZj in (2) toobtainv2u =2kT jZj jH 0j21 + cos'(3)which is different from that given by [1, eq. (6)] [derived from (1)],whose dimensions were not those of a voltage spectral density and in-cludedH 0(s) instead of jH 0j. Nevertheless, the conclusion in [1] is cor-rect; the output voltage spectral density will increase if the impedanceZ is not purely resistive.This interesting result has no simple explanation and deserves fur-ther elaboration because it might lead to the wrong conclusion that areactive impedance Z would increase the output noise. According to(3), the voltage spectral density at the output of an amplifier whoseinput impedance includes, say, a shunting capacitance, will certainlyincrease when R = jZj, but, because of the frequency dependence ofthe impedance of a capacitance, that condition will happen at a singlefrequency. What will then happen in a finite frequency bandwidth? IfZ = R + jX, the output noise power from fL to fH will beV 2u =ffv2u df=4kTRffZR+ Z2jH 0j2 df=4kTRffR 2 +X 2(R+R)2 +X 2jH 0j2 df (4)whose result is determined by the frequency dependence of R, X,and H 0. For the sake of simplicity, let us assume that jH 0j = 1. For aManuscript received March 21, 2002; revised March 21, 2002.The authors are with the Departament d’Enginyeria Electrònica, UniversitatPolitècnica de Catalunya, Barecelona, Spain (e-mail: elerpa@eel.upc.es).Publisher Item Identifier S 0018-9456(02)04873-8.Fig. 1. Effect of the thermal noise of a resistor in a circuit can be analyzed byseparating the resistor from the rest of that circuit.purely capacitive input impedance (R = 0, 1=X =  2fC), wewould haveV 2u R =0 =kTC2arctan2(fH   fL)RC1 + 2pfHfLRC2 (5)whose maximal value would beV 2u(max) R =0 = 4kTR fLfH arctanfH   fL2pfHfL(6)atR = Z j2 fLfH =12pfLfHC: (7)For a purely resistive input impedance (Z = R), we would haveV 2u Z=R = 4kTRRR+R2(fH   fL) (8)whose maximal value would beV 2u(max) Z=R = kTR(fH   fL) (9)at R = R.Dividing (6) by (9) yieldsV 2u(max) R =0V 2u(max)Z=R=4pfLfHfH   fL arctanfH   fL2pfHfL: (10)When fH   fL pfHfL, the output noise power ratio is 2. WhenfH   fL pfHfL, the output noise power ratio isV 2u(max) R =0V 2u(max)Z=R= 4pfLfHfH   fL2=2pfLfHfH   fL  1: (11)Therefore, from the point of view of output noise, when the source re-sistance and the input impedance are matched (R = jZj), for narrow-band signals it is advisable to reduce the input capacitance as much aspossible, whereas for a wide band signal, the input capacitance helpsin reducing the output noise power. In the usual cases of voltage or cur-rent measurement we haveR jZj andR jZj, respectively, whichresults in minimal output voltage noise spectral density. By the sametoken, the output noise contributed by the amplifier input resistanceRwill be small whenever it is unmatched toR. Thus, the common designpractice to reduce voltage or current loading also reduces output noise.REFERENCES[1] R. Giannetti, “On resistor-induced thermal noise in linear circuits,”IEEE Trans. Instrum. Meas., vol. 49, pp. 87–88, Feb. 2000.0018-9456/02$17.00 © 2002 IEEE

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